reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;
reserve F,F1,F2,F3 for Functor of A,B,
  G,G1,G2,G3 for Functor of B,C,
  H,H1,H2 for Functor of C,D,
  s for natural_transformation of F1,F2,
  s9 for natural_transformation of F2,F3,
  t for natural_transformation of G1,G2,
  t9 for natural_transformation of G2,G3,
  u for natural_transformation of H1,H2;

theorem Th26:
  G1 is_naturally_transformable_to G2 & G2
  is_naturally_transformable_to G3 implies (t9`*`t)*F = (t9*F)`*`(t*F)
proof
  assume
A1: G1 is_naturally_transformable_to G2 & G2 is_naturally_transformable_to G3;
  then
A2: G1*F is_naturally_transformable_to G2*F & G2*F
  is_naturally_transformable_to G3*F by Th20;
  now
    let a be Object of A;
A3: G1.(F.a) = (G1*F).a & G2.(F.a) = (G2*F).a by CAT_1:76;
A4: G3.(F.a) = (G3*F).a by CAT_1:76;
A5: t9.(F.a) = (t9*F).a & t.(F.a) = (t*F).a by A1,Th22;
    thus ((t9`*`t)*F).a = (t9`*`t).(F.a) by A1,Th22,NATTRA_1:23
      .= (t9.(F.a))*(t.(F.a)) by A1,NATTRA_1:25
      .= ((t9*F)`*`(t*F)).a by A2,A5,A3,A4,NATTRA_1:25;
  end;
  hence thesis by A2,Th24,NATTRA_1:23;
end;
